Homotopy in Exact Categories

Abstract

In this monograph we develop various aspects of the homotopy theory of exact categories. We introduce different notions of compactness and generation in exact categories $E$, and use these to study model structures on categories of chain complexes $Ch_{*}(E)$ which are induced by cotorsion pairs on $E$. As a special case we show that under very general conditions the categories $Ch_{+}(E)$, $Ch_{\ge0}(E)$, and $Ch(E)$ are equipped with the projective model structure, and that a generalisation of the Dold-Kan correspondence holds. We also establish conditions under which categories of filtered objects in exact categories are equipped with natural model structures. When $E$ is monoidal we also examine when these model structures are monoidal and conclude by studying some homotopical algebra in such categories. In particular we provide conditions under which $Ch(E)$ and $Ch_{\ge0}(E)$ are homotopical algebra contexts, thus making them suitable settings for derived geometry.